arXiv research

Study counts geodesics on hyperbolic 3-manifolds, proving prime theorems.

3-manifolds are chiral if not finitely covered by sphere or product.

Computes virtually cyclic dimension for 3-manifold groups.

This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of view.

Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.

The paper extends monodromy theory to incompressible surfaces in 3-manifolds and applies it to link primeness.

Finite covers of 3-manifolds have non-trivial torsion in their first homology.

An open book decomposition of a 3-manifold $M$ induces a Heegaard splitting for $M$, and the minimal genus among all Heegaard splittings induced by open book decompositions is called the \emph{open book genus} of $M$. It is conjectured by Ozbagci \cite{O} that the open book genus is additive under the connected sum of …

Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the r…

Study the moduli space of reducible 3-manifolds using prime decomposition.

We establish an existence and uniqueness theorem for prime decompositions of theta-curves in $3$-manifolds.

Researchers calculate the minimal entropy of 3-manifolds, proving it's additive.

Study on geodesics on connected sums of manifolds, showing infinite number of distinct closed geodesics.

Simplified proof classifies surfaces using normal curves.

Study shows complexity bounds for prime 3-manifolds using cohomology.

The absolute Galois group of 3-manifolds determines their structure up to homeomorphism.

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These inv…

The chromatic number of sphere graphs in 3-manifolds is bounded.

3-manifolds' volumes match stable integral values.

We developed an efficient algorithm to factorize knots.

The paper presents new algebraic structures on the 2-sphere using topological field theories.

3-manifolds study Hasse norm principle, akin to number fields.

Homogeneous braids are visually prime, solving a Cromwell question.

In this paper we use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links to commute under the stacking operation.

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples.…

It has been shown by V. Colin that every tight contact 3-manifold can be written as a connected sum of prime manifolds. Here we prove that the summands in this decomposition are unique up to order and contactomorphism.

Paper shows unique decomposition of 3-manifolds and multiplicative property of Reidemeister torsion.

We improve and extend to the non-orientable case a recent result of Karabas, Malicki and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.

This work classifies belted sum decompositions of fully augmented links.

We classify those compact 3-manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold $M$ is prime and orientable and the fundamental group of $M$ is non-trivial then $M \cong Σ\times S^1$, where $Σ$ is a surface.

We give criteria for framed links and 3-manifolds to be periodic of prime order. As applications we show that the Poincare sphere is of periodicity 2, 3, 5 only and the Brieskorn sphere (2,3,7) is of periodicity 2, 3, 7 only.

Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.

The Frohman Kania-Bartoszynska ideal is an invariant associated to a 3-manifold with boundary and a prime p >3. We give some estimates of this ideal. We also calculate this invariant for some 3-manifolds constructed by doing surgery on a knot in the complement of another knot.

We prove hyperbolicity for links in thickened surfaces.

Proves regularity of isomorphisms between hyperbolic 3-manifolds.

Study open 3-manifolds as sums of closed ones, finding a classification.

We show that for any set of primes $\mathcal{P}$ there exists a space $M_{\mathcal{P}}$ which is a homology and cohomology 3-manifold with coefficients in $\mathbb{Z}_{p}$ for $p\in \mathcal{ P}$ and is not a homology or cohomology 3-manifold with coefficients in $\mathbb{Z}_q$ for $q\not\in \mathcal{ P}$. Moreover, $M…

The paper solves CR curvature prescription on pseudo-Einstein 3-manifolds.

We show that a hyperbolic $3$-manifold can be the cyclic branched cover of at most fifteen knots in $\mathbf{S}^3$. This is a consequence of a general result about finite groups of orientation preserving diffeomorphisms acting on $3$-manifolds. A similar, although weaker, result holds for arbitrary irreducible $3$-mani…

This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3-manifolds. In particular, it is shown that the p-adic valuation of the quantum SO(3)-invariant of a 3-manifold M, for odd primes p, is bounded below by a linear function of the mod p first betti number of M. Shar…

The study proves representations for certain 3-manifolds using gauge theory.

We construct a simple topological invariant of certain 3-manifolds, including quotients of the 3-sphere by finite groups, based on the fact that the tangent bundle of an orientable 3-manifold is trivialisable. This invariant is strong enough to yield the classification of lens spaces of odd, prime order. We also use pr…

Unified framework for Alexandrov 3-spaces, extending manifold results.

This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any 3-manifold $M$, unless $N$ is finitely covered by either a torus bundle, or a trivial circle…

Templates are branched 2-manifolds with semi-flows used to model `chaotic' hyperbolic invariant sets of flows on 3-manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be boun…

Let $M$ be a closed, oriented, connected 3--manifold and $(B,π)$ an open book decomposition on $M$ with page $Σ$ and monodromy $\varphi$. It is easy to see that the first Betti number of $Σ$ is bounded below by the number of $S^2\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is…

Knots in 3-manifolds are equivalent if isotopic, except in special cases.

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures: A reducible 3-manifold admits an Anosov torus if an…